Problem: Evaluate the definite integral. $\int^{12}_{3}\big(6x+1\big)\,dx = $
Solution: First, use the power rule: $\int^{12}_{3}\big(6x+1\big)\,dx ~=~(3 x^2+x)\Bigg|^{{12}}_{3}$ Second, plug in the limits of integration: $\big[3\cdot{12}^2+{12}\big]-\big[3\cdot{3}^2+{3}\big] = 444 - 30 = 414$. The answer: $\int^{12}_{3}\big(6x+1\big)\,dx ~=~ 414$